Effect of Partial-Band Interference on
Receiver Estimation of C/N0: Theory
John W. Betz, The MITRE Corporation
BIOGRAPHY
John W. Betz is a Consulting Engineer at The MITRE
Corporation. He received a Ph.D. in Electrical and
Computer Engineering from Northeastern University. His
work involves development and analysis of signal
processing for communications, navigation, radar, and other
applications. He developed the Binary Offset Carrier
modulation for the GPS M code signal, and more recently
has been contributing to theoretical predictions of receiver
performance. During 1998 and 1999, he led the GPS
Modernization Signal Design Team’s Modulation and
Acquisition Design Subteam, contributing to many aspects
of the signal design and evaluation. He has authored many
technical papers and reports on theory and applications of
signal processing.
ABSTRACT
Testing the response of C/A code receivers to partial-band
interference (continuous Gaussian interference whose
power is concentrated in part of the front-end bandwidth)
has included examining how receivers measure and report
the effect of such interference. This paper considers two
fundamentally different ways that receivers measure and
report the effect of interference on signal quality,
demonstrating that in non-white noise the measures are not
the same. While the effective C / N0 reliably measures the
effect of interference on a receiver, the precorrelation
C / N0 is not reliable. Specifically, precorrelation
estimation of N0 does not properly account for the
spectrum of the interference. Depending on the spectrum of
the interference, precorrelation estimates may be accurate,
or may overestimate the degradation caused by interference,
or may underestimate the degradation caused by
interference. Theoretical and numerical results are provided
and compared to some measured data.
INTRODUCTION
Many receivers report received signal quality as carrierpower-to-noise density ratio, denoted C / N0 . Testing
reported in a companion paper [1] has found that different
receivers report different C / N0 when presented with the
same non-white interference. These different responses
cannot be explained solely by differences in front-end
bandwidths, discriminator designs, or other receiver
characteristics. This paper suggests how such curious
results may occur, identifies an objectively correct measure
of interference effects on signal quality, shows which
estimate of C / N0 corresponds to the correct measure, and
provides theoretical predictions of the response to different
interference spectra.
It has long been recognized that spread spectrum receivers
mitigate narrowband interference—the term “processing
gain” has been used to describe and quantify this capability.
Analysis describing this behavior is available in many
standard references. The underlying mathematical
development in [2] defines two different but related
quantities: coherent output signal-to-noise-plus-interference
ratio (SNIR), and noncoherent SNIR. These output SNIRs
are also known as postcorrelation SNIRs, since they are
defined at the output of the crosscorrelation between
reference signal and received signal. Postcorrelation
quantities occur after the entire correlation operation (both
multiplication by the reference and integration by a time
much longer than the chip period).
Coherent output SNIR is defined under the assumption that
the phase of the reference signal is aligned with that of the
received signal. The estimates of carrier phase and
frequency used for carrier tracking and message
demodulation are derived from the sequence of inphase
parts of the crosscorrelation, computed using time segments
of reference and received signal. Thus, the coherent output
SNIR characterizes how well the receiver tracks carrier
frequency and phase, and demodulates the data message
that is phase shift key modulated onto the spread carrier.
Noncoherent output SNIR is defined under the assumption
that the phase of the reference signal has unknown
alignment with that of the received signal. The test statistic
used in acquisition is derived from the sum of squared
inphase and quadrature parts of the crosscorrelation. Thus,
the noncoherent output SNIR characterizes how well the
receiver acquires the signal.
Neither output SNIR measure predicts code tracking
performance in non-white interference. A separate measure,
presented in [3, 4], describes code tracking performance.
Although the coherent output SNIR and noncoherent SNIR
are distinct quantities, [2] shows that there is a single
fundamental quantity, termed the effective C / N0 , that is
related to both of them. When there is no non-white
interference, effective C / N0 is simply the conventional
C / N0 . When non-white interference is present, effective
C / N0 reliably describes how the combination of partialband interference and noise affect both coherent output
SNIR and noncoherent output SNIR, hence how it affects
the receiver functions of carrier tracking, data
demodulation, and acquisition. It is thus a uniquely relevant
descriptor of how interference affects signal quality.
The analysis in this paper focuses on the essence of C / N0
estimation under idealized conditions. Multiple access
interference, especially crosscorrelations between signals,
are not considered. Neither are channel effects such as
multipath. Finally, the bandwidth of the interference is
assumed to be much wider than the reciprocal of the
integration time used in the correlator, and the integration
time is much greater than the period of a spreading symbol,
or chip.
Many receivers estimate and report some measure of the
received signal quality for each signal being tracked,
typically calling this signal quality estimate a measurement
of C / N0 . Based in part on [5], this paper describes two
distinct ways that receivers may estimate C / N0 . It shows
that both estimates are equivalent when the interference has
a flat spectrum over the receiver front-end, but that they
differ when non-white interference is present. In non-white
interference, one of the estimates provides a valid measure
of effective C / N0 , while the other cannot be related to
effective C / N0 . In fact, this latter measure may either
grossly overestimate or grossly underestimate the effective
C / N0 and thus the received signal quality, depending on
the spectrum of the interference.
The next section summarizes theoretical results from [2]
that are the foundation of this paper. The following section
describes two different estimates of C / N0 that may be
used in receivers. Next, the behavior of these estimates in
the presence of non-white interference is analyzed, and
expressions are derived for some limiting cases. Numerical
results are used to contrast the two estimates, and
predictions are compared to measured behavior of actual
receivers. Finally, results and conclusions are summarized.
frequency of arrival is known to within a fraction of the
reciprocal of the coherent integration time, this assumption
applies. The received data is then the sum of the signal and
noise plus interference, w(t ) . The received data is observed
over
a
long
time
interval
Tobs :
x (t ) = exp{iθ }s(t − t0 ) + w(t ) , 0 ≤ t ≤ Tobs , where the
unknown delay, or time of arrival, t0 , is defined relative to
an arbitrary origin. The received noise plus interference is
statistically independent of the signal.
Consistent with most analyses of code tracking accuracy, it
is assumed that the time of arrival, t0 , is fixed, so no
dynamics are considered explicitly. However, the results
hold as long as any change in t0 is somewhat slower than
the reciprocal bandwidth of the code-tracking loop. Without
loss of generality, t0 is set to zero.
The signal of interest (henceforth called the signal) is s(t ) ;
all signals not of interest, plus thermal noise, are grouped
into w(t ) . The analysis in this paper assumes that w(t ) is
continuous, Gaussian, stationary, and circularly symmetric.
It is assumed that any automatic gain control (AGC) in the
receiver responds slowly compared to the bandwidths of
signal, noise, and interference, so the gain over the time of
interest is approximated by a constant γ . The front end of
the receiver, including mixing and quantization, is modeled
as linear. Also, the transfer function of the receiver chain is
approximated by a constant within complex bandwidth β r ,
and zero outside that bandwidth.
The signal has power spectrum CGs ( f ) , where Gs ( f ) is
the power spectral density normalized to unit area over
infinite limits, and C is the received power of the signal.
The thermal noise has power spectral density N0 , and the
interference has power spectral density Cι Gι ( f ) with
THEORETICAL BACKGROUND
When the front-end bandwidth is wide enough to contain
essentially all of the signal power, (1) becomes
This analysis assumes a (complex-valued) baseband signal
s(t ) that is known (except for unknown delay and perhaps
phase) at the receiver. While it is assumed that the
frequency of arrival is known perfectly, as long as the
β /2
∫− βr r / 2 Gι ( f )df = 1 . Assume that the gain in the receiver
processing chain is known, and let the integration time for
correlating received data against reference signal be T .
[2] derives the mean and variance of the real part of the
crosscorrelation between the received data and the
reference signal. Coherent output SNIR is the squared mean
when the reference signal is aligned in delay, frequency,
and phase, divided by the variance, (which is shown in [2]
to be independent of the delay):
2

C  βr / 2
 ∫ Gs ( f )df 
N0 − β r / 2


ρc = β / 2
.
βr / 2
r
Cι
∫ Gs ( f )df +
∫ Gι ( f )Gs ( f )df
N0 − β r / 2
−βr / 2
2T
(1)
2T
ρc =
C
N0
βr / 2
,
Cι β r / 2
1+
∫ Gι ( f )Gs ( f )df
N0 − β r / 2
∫
−βr / 2
Gs ( f )df = 1. (2)
Similarly, [2] derives the mean and variance of the quantity
resulting from taking the magnitude squared of the
crosscorrelation between the received data and the
reference signal. Noncoherent output SNIR is the
magnitude-squared mean of this quantity when the
reference signal is matched in delay and frequency, but with
arbitrary phase alignment, divided by its variance, (which is
shown in [2] to be independent of the delay):
2

C  βr / 2
T
 ∫ Gs ( f )df 
N0 − β r / 2


ρn = β / 2
+ 1.
β
/
2
r
Cι r
∫ Gs ( f )df +
∫ Gι ( f )Gs ( f )df
N0 − β r / 2
−βr / 2
(3)
The effective C / N0 is defined as
βr / 2
 C 
 =

 N0  eff
−βr / 2
βr / 2
N0
∫
−βr / 2
Gs ( f )df + Cι
∫
−βr / 2
βr / 2
(5)
−1
βr / 2


G
f
G
f
df
(
)
(
)


∫
ι
s
Cι − β r / 2
C 

=
+
1
 .
βr / 2
N0  N0

∫ Gs ( f )df 


−βr / 2


C
N0 + Cι
βr / 2
∫
−βr / 2
Gι ( f )Gs ( f )df
(6)
−1
=
βr / 2

C βr / 2
C 
1 + ι ∫ Gι ( f )Gs ( f )df  , ∫ Gs ( f )df = 1.
N0  N0 − β r / 2
 − β r / 2

β /2
∫ Gs ( f )df = 1.
(8)
−β / 2
Similarly, comparing (5) with (3) yields the noncoherent
output SNIR
 C  βr / 2
ρn = T 
∫ Gs ( f )df + 1,

 N0  eff − β r / 2
(9)
βr / 2
∫
−βr / 2
Gs ( f )df = 1.
(10)
Consequently, the effective C / N0 is sufficient for
describing how interference affects both the coherent and
noncoherent output SNIRs, hence carrier tracking, data
message demodulation, and acquisition.
[2] also shows that the variance of the coherent correlator
output at any lag value is proportional to a quantity that will
be called the postcorrelation noise density:
( N0 )post = N0
When the front-end bandwidth is wide enough to contain
essentially all of the signal power, (5) becomes
 C 

 =
 N0  eff
 C 
ρ c = 2T 
 ,
 N0  eff
 C 
ρn = T 
 + 1,
 N0  eff
Gι ( f )Gs ( f )df
C
∫ Gs ( f )df
N0 − β r / 2
= β /2
r
Cι β r / 2
∫ Gs ( f )df +
∫ Gι ( f )Gs ( f )df
N0 − β r / 2
−βr / 2
(7)
or, when the front-end bandwidth is wide enough to contain
essentially all of the signal power,
Gs ( f )df
βr / 2
 C  β /2
ρ c = 2T 
∫ Gs ( f )df ,

 N0  eff − β / 2
or, when the front-end bandwidth contains essentially all
the signal power,
When the front-end bandwidth is wide enough to contain
essentially all of the signal power, (3) becomes
C
T
βr / 2
N0
1
ρn =
+
,
∫ Gs ( f )df = 1. (4)
Cι β r / 2
−βr / 2
1+
∫ Gι ( f )Gs ( f )df
N0 − β r / 2
C ∫
(5) and (6) show how interference affects the effective
C / N0 . As expected, only interference passed by the
receiver front end has any effect. However, multiplication
of the interference spectrum by the signal spectrum within
the integral shows that the interference also is effectively
filtered with magnitude-squared transfer function the power
spectrum of the signal. Thus, if the signal has much
narrower bandwidth than the front-end bandwidth,
interference outside the signal bandwidth but within the
front-end bandwidth has substantially less effect on
effective C / N0 than interference having the same power
near band center.
Comparing (5) with (1) shows that the coherent output
SNIR is
βr / 2
∫
−βr / 2
Gs ( f )df + Cι
βr / 2
∫
−βr / 2
Gι ( f )Gs ( f )df . (11)
The numerator in the first line of (5) is the received signal
power passed through the front end (since the received
signal power C is defined over infinite bandwidth, and the
signal’s power spectrum Gs ( f ) is normalized to unit area
over infinite bandwidth). Further, the postcorrelation noise
density (11) is the same as the denominator in the first line
of (5), indicating that the postcorrelation noise density is the
noise density that affects the effective C / N0 . In (11), the
interference is effectively filtered by the signal spectrum, as
described in the discussion after (6).
When the front-end bandwidth is wide enough to contain
essentially all of the signal power, (11) becomes
( N0 )post = N0 + Cι
βr / 2
∫
−βr / 2
βr / 2
∫
−βr / 2
Gι ( f )Gs ( f )df ,
(12)
and the average of the real part of K of them, each with the
references aligned in time,

1 K
Ĉ =  ∑ ℜ{ck (0)} ,
(16)
K  k =1

Gs ( f )df = 1 .
Consider the precorrelation noise density:
( N0 )pre =
Cι β r / 2
N0 β r / 2
∫ Gι ( f )df
∫ df +
βr −βr / 2
βr −βr / 2
C βr / 2
= N0 + ι ∫ Gι ( f )df .
βr −βr / 2
(13)
While this quantity does not occur in any of the expressions
that describe how interference affects receiver performance,
it arises in the next section. (13) is similar to (11), except
that in (13), the noise and interference spectra are not
multiplied by the signal spectrum. When all of interference
power is within the front-end bandwidth, (13) simplifies to
C
( N0 )pre = N0 + β ι ,
r
βr / 2
∫
−βr / 2
Gι ( f )df = 1 ,
(14)
showing that the interference power is reduced by the frontend bandwidth, independent of interference spectral shape.
RECEIVER ESTIMATION OF INTERFERENCE
EFFECTS
This section describes two possible approaches for
estimating the effect of interference in a receiver. While
many ad hoc approaches could be defined, the previous
section demonstrates that the effective C / N0 defined in
(3) uniquely describes the effect of interference from the
point of acquisition performance, carrier tracking
performance, and message demodulation performance.
Thus, an estimate provides a meaningful measure of
interference effects if it approximates the effective C / N0 .
Since the measure should apply when the signal carrier
power is much less than the power in the noise plus
interference at the front end, the estimate of signal power
must be based on the correlation output, so that the signal
power is enhanced relative to the noise and interference.
This estimation of signal power is described in the first
subsection. Two different approaches for estimating the
effective noise density, however, are considered in
subsequent subsections. While different specific algorithms
may be employed in receivers, it is expected that their
behavior is consistent with one of these two approaches.
Estimation of Signal Power
Denote the received data after front-end filtering by
x1 (t ) = exp{iθ }s1 (t ) + w1 (t ) , and assume that the front-end
β /2
reference signal r (t ) is aligned with the carrier phase and
frequency, the k th correlator output is
1 ( k +1)T
*
− iθ
ck (τ ) =
(15)
∫ x1 (t )r (t − τ )e dt,
T kT
bandwidth is large so that ∫− βr / 2 Gs ( f )df ≅ 1 . When the
r
approaches the signal power when K is large.
Estimation of Postcorrelation Noise Density
As seen in the analysis in [2], the variance of ck (τ ) is twice
the variance of either ℜ{ck (τ )} or ℑ{ck (τ )} , because of
the assumption that noise plus interference is Gaussian and
circularly symmetric. None of these variances depend on τ .
The sample variance of ℜ{ck (τ )} is then
2
 
1 K 
1 K
 ∑  ℜ{ck (τ )} −
∑ ℜ{cl (τ )}  . The estimation
K  k =1 
K l =1
 


process can be simplified by choosing a value of τ at least
several chip periods away from the correlation peak, so that

1 K
∑ ℜ{ck (τ )} ≅ 0 and the sample variance is then
K k =1

approximated by the average squared value
2
1 K
∑ ℜ{ck (τ )}  when K is large. Since the average
K k =1

value of ℑ{ck (τ )} is zero for all values of τ , an alternate
(
)
2
1 K
∑ ℑ{ck (τ )}  for
K k =1

any value of τ . In fact, variance estimation can employ the
prompt correlator output at the orthogonal phase:
2
1 K
∑ ℑ{ck (0)}  , again avoiding the need to subtract

K  k =1

the average squared value, and also using correlator taps
that are already available in a typical code tracking loop.
Note that these estimates use only the real part or only the
imaginary part of the correlator outputs, implying that the
receiver is accurately tracking carrier phase.
When K is large and ergodicity applies, these estimates
approach the variance of ℜ{ck (τ )} , which [2] shows is
βr / 2

C 
Var ℜ{ck (τ )} =
 N0 + Cι ∫ Gι ( f )Gs ( f )df . (17)
2T 

−βr / 2

estimate of the sample variance is
(
{
(
)
)
}
Thus, there are several equivalent estimates of the
postcorrelation noise density
( )post
Nˆ 0
=
2
K
 
2T 1  K 
 ∑  ℜ{ck (τ )} − ∑ ℜ{cl (τ )} , any τ
 
Cˆ K k =1 
l =1

=
2
2T 1  K
 ∑ ℜ{ck (τ )} ,
Cˆ K k =1

any τ away from peak
=
2
2T 1  K
 ∑ ℑ{ck (τ )} ,
ˆ
K
C
 k =1

any τ , including τ = 0.
(
)
(
)
(24)
(18)
As long as K is large, and assuming that C is known, the
postcorrelation noise density estimate approaches its mean
βr / 2
( )post  = N0 + Cι −β∫ / 2 Gι ( f )Gs ( f )df .

E  Nˆ 0

(19)
r
As long as K is large, dividing (16) by (18) yields the
postcorrelation C / N0 , which is a close approximation to
the effective C / N0 .
Estimation of Precorrelation Noise Density
Suppose instead that the receiver estimates the power in the
1 LT
2
received data after front-end filtering by
∫ x1 (t ) dt ,
LT 0
then forms the precorrelation noise density estimate by
dividing this power estimate by the front-end bandwidth
1 LT
2
Nˆ 0
=
(20)
∫ x1 (t ) dt.
pre
β r LT 0
( )
In (20), L is the number of correlation integration times
2
used to form the estimate. The mean of x1 (t ) is
{
E x1 (t )
2
} = E { s (t )
1
βr / 2
=C ∫
−βr / 2
2
+ w1 (t )
2
Gs ( f )df +
}
βr / 2
∫
−βr / 2
[ N0 + Cι Gι ( f )]df .
(21)
As long as the signal power at the input is much less than
the combined power of noise and interference at the input,
(21) is approximated by
{
E x1 (t )
2
}
≅
βr / 2
∫
−βr / 2
[ N0 + Cι Gι ( f )]df , low input SNIR.
(22)
As long as L is large, the precorrelation noise density
estimate approaches its mean
1 LT
2


E  Nˆ 0
=
∫ E x1 (t ) dt

pre  β r LT 0

{
( )
≅
}
1 βr / 2
∫ N0 + Cι Gι ( f ) df
βr −βr / 2
[
Dividing (16) by (23) yields an estimate of the
precorrelation C / N0 .
βr / 2




∫ Gs ( f )df
 C 
C 
−βr / 2

=


, low input SNIR.
βr / 2
 N0  pre N0 
C
1
ι
1 +
∫ Gι ( f )df 
N0 β r − β r / 2




]
(23)
C βr / 2
= N0 + ι ∫ Gι ( f )df , low input SNIR.
βr −βr / 2
Observe that (23) is the same as the precorrelation noise
density defined in (13).
When the front-end bandwidth is wide enough to contain
essentially all of the signal power,
−1

 C 
Cι 1 β r / 2
C 
=
1 +
∫ Gι ( f )df  ,


 N0  pre N0  N0 β r − β r / 2

β /2
(25)
low input SNIR, ∫ Gs ( f )df = 1.
−β / 2
While there are many ways of estimating precorrelation
noise density, one pointed out in [5] merits special
consideration. While estimating the precorrelation noise
density can be performed directly using power
measurement circuitry or an equivalent algorithm on digital
samples, a simpler alternative may exist using AGC
circuitry. AGC circuitry monitors the input power (for
example, by counting the fraction of time that the most
significant bit is triggered) to control the RF gain. Since this
feedback signal is related to input power, it can be used
(with suitable calibration) to estimate the precorrelation
noise density.
Comparison of Estimates
Observe that both the estimate of postcorrelation C / N0
and the estimate of precorrelation C / N0 use the same
postcorrelation estimate of C . The essential difference
between them is the estimate of N0 used—the
postcorrelation estimate of N0 has the same form as the
denominator in effective C / N0 , with the signal’s spectrum
filtering the interference spectrum, while the precorrelation
estimate of N0 fundamentally differs from any known
expression for receiver performance, since the spectral
shape of the interference within the front-end bandwidth
has no effect on the estimate.
Since the postcorrelation C / N0 is a close approximation to
the effective C / N0 , the remaining discussion treats the
two as identical, and calls both quantities the effective
C / N0 . The precorrelation C / N0 is different, so it is
considered separately.
Since the expressions are simpler when the front-end
bandwidth is wide enough to contain essentially all of the
signal power and all of the interference power, this is
assumed throughout the remainder of this section.
However, the results would be qualitatively similar if this
assumption were not applied.
If the interference has flat spectrum over the front-end
bandwidth,
β
1
 , f ≤ r
Gι ( f ) =  β r
(26)
2
 0, otherwise,
with total power Cι , the effective C / N0 with this
interference is obtained by substituting (26) into (6),
yielding
−1
βr / 2
 C 
Cι 1 
C 
1 +
 ,

 =
 N0  eff N0  N0 β r 
∫
−βr / 2
Gs ( f )df = 1.
(27)
Similarly, the precorrelation C / N0 is obtained by
substituting (26) into (25), yielding
 C 
C
=


 N0  pre N0
−1

Cι 1 
1 +
 ,
N
0 βr 

(28)
β /2
low input SNIR, ∫ Gs ( f )df = 1.
−β / 2
−1
β /2
∫ Gs ( f )df = 1
(29)
−β / 2
with total power Cι and βι much less than the chip rate of
the signal so that Gs ( f ) is approximately constant over βι .
Then the effective C / N0 for interference at band center is
obtained by substituting (30) into (6), yielding
−1
C
N0
(32)
β /2

Cι 1 
1 +
 , low input SNIR, ∫ Gs ( f )df = 1.
N
0 βr 

−β / 2
Observe that while the expressions for effective C / N0 and
precorrelation C / N0 have similar form (29), the constants
multiplying the interference power in the denominator are
not the same. To explore this difference further, let the
signal have a conventional BPSK modulation with chip
period Tc the reciprocal of the chip rate fc ,
Gs ( f ) = Tcsinc 2 [πfTc ].
(33)
This analysis neglects any spectral lines arising from
periodic spreading codes. As long as the interference
bandwidth is much greater than any such period, this is a
valid approximation.
Then Gs (0) = Tc = 1 / fc , and (31) becomes
 C 
Cι 1 
C 
1 +
 ,
 ≅

 N0  eff N0  N0 fc 
βr / 2
∫
−βr / 2
Gs ( f )df = 1.
(34)
This shows that the interference power degrades effective
C / N0 diminished by the processing gain of the signal,
where the processing gain is equal to the chip rate
fc = 1 / Tc . This is a well-known result that describes the
performance of a spread spectrum receiver in the presence
of narrowband interference at band center.
β /2
where Γ is a constant that turns out to differ for the
effective C / N0 and precorrelation C / N0 .
Assume that the interference occupies a narrow bandwidth
directly in band center, for example,
β
1
f ≤ ι
 ,
Gι ( f ) =  βι
(30)
2
 0, otherwise,

 C 
Cι 1 βι / 2
C 

=
+
1
∫ Gs ( f )df 


 N0  eff N0  N0 βι − βι / 2

=
−1
−1
(27) and (28) show that precorrelation C / N0 is the same
as effective C / N0 for interference with flat spectrum.
When the interference occupies a narrow bandwidth (much
less than the chip rate but much larger than any periodicity
in the spreading sequence), both the effective C / N0 and
precorrelation C / N0 are approximately of the form
 C 
Cι 
C 
Γ ,
1 +
=

 N0  N0  N0 
−1
 C 
Cι 1 βι / 2 
C 
1 +
=
∫ df 


 N0  pre N0  N0 β r βι − βι / 2 
−1
(31)
βr / 2

Cι
C 
≅
Gs (0) ,
∫ Gs ( f )df = 1.
1 +
N0  N0

−βr / 2
In contrast, substituting (30) into (25) yields the
precorrelation C / N0 for this case:
For ∫− βr / 2 Gs ( f )df = 1 to be a valid approximation, the
r
front-end bandwidth must be much greater than the chip
rate, so that β r Tc >> 1 . Then the interference power is
multiplied by a much smaller number in the precorrelation
C / N0 than in effective C / N0 , meaning that, as
interference power increases, its effect is not fully reported
by precorrelation C / N0 .
For example, suppose that the signal spectrum is given by
(33), and β r = 8 fc , equivalent to a front-end bandwidth of
approximately 8 MHz for a C/A code receiver. Then the
precorrelation C / N0 reports less degradation of effective
C / N0 than actually occurs—here it underestimates the
effect of the interference by approximately 9 dB. The
receiver is actually degraded as predicted by effective
C / N0 , but the precorrelation C / N0 does not accurately
reflect this degradation.
Now suppose the interference power is concentrated near
the edge of the front-end bandwidth. As long as the signal
spectrum is symmetric, it makes no difference if the
interference spectrum is symmetric, with power split
between both band edges, or if all the interference power is
near one band edge. For example, let
1
 ,
Gι ( f ) =  βι
 0,
β
f − fι ≤ ι
2
otherwise,
(35)
NUMERICAL RESULTS
with
total
power
and
Cι
− β r / 2 ≤ fι − βι / 2 < fι + βι / 2 ≤ β r / 2 . The effective
C / N0 with interference concentrated near fι is obtained
by substituting (35) into (6), yielding

 C 
Cι 1 fι + βι / 2
C 

=
+
1
∫ Gs ( f )df 


 N0  eff N0  N0 βι fι − βι / 2

−1
−1
(36)
βr / 2

Cι
C 
≅
Gs ( fι ) ,
∫ Gs ( f )df = 1.
1 +
N0  N0

−βr / 2
In contrast, substituting (35) into (26) yields
 C 
C
=


 N0  pre N0
about these aspects of receiver performance when the
interference spectrum is not flat.
−1

Cι 1 
1 +
 ,
 N0 β r 
For computation of numerical results, consider a C/A code
receiver with thermal noise density –203 dBW/Hz and
received signal power of –158 dBW, for a nominal received
C / N0 of 45 dB-Hz, neglecting implementation losses.
The first results consider interference at band center with a
bandwidth of 10 kHz. Figure 1 compares effective C / N0
and precorrelation C / N0 for a receiver front-end
bandwidth of 8 MHz. While effective C / N0 relates the
true effect of interference on acquisition, carrier tracking,
and data message demodulation, precorrelation C / N0 does
not indicate the full degrading effect of this interference,
underestimating the effect of the interference by
approximately 9 dB as predicted in the previous section.
46
(37)
β /2
44
low input SNIR, ∫ Gs ( f )df = 1.
42
−β / 2
 C 
C

 ≅
 N0  eff N0
βr / 2
∫
−βr / 2
−1


Cι
1
1 +
 ,
2
 N0 (3.5π ) fc 
(38)
Gs ( f )df = 1,
while (37) becomes
−1
 C 
Cι 1 
C 
=
1 +
 ,


N
N
N
 0  pre
0 
0 8 fc 
β /2
C/N0 (dB-Hz)
40
Effective C / N0 depends on the signal spectrum evaluated
at the center frequency of the interference, while the
precorrelation C / N0 depends only on the front-end
bandwidth, and not on the signal spectrum nor on the
spectrum of the interference. In fact, (37) is identical to the
result (32) obtained for interference at band center.
If the signal spectrum is given by (33), β r = 8 fc and
fι = 3.5 fc , interference is near band edge so (36) becomes
38
36
34
32
30
-160
Effective C/N0
Precorrelation C/N0
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 1. Theoretical Results for Narrowband Interference
at Band Center, Receiver Front-End Bandwidth 8 MHz
Figure 2 shows the same quantities as in Figure 1, but for a
receiver with 16 MHz front-end bandwidth. For this widerbandwidth receiver, precorrelation C / N0 underestimates
the effect of the interference by approximately 12 dB.
While the receiver is significantly degraded as the
interference power exceeds –135 dBW, the precorrelation
C / N0 only reports mild degradation.
46
(39)
44
low input SNIR, ∫ Gs ( f )df = 1.
42
−β / 2
C/N0 (dB-Hz)
40
Here, the precorrelation C / N0 reports more degradation of
effective C / N0 than actually occurs—for this example it
overestimates the effect of the interference by
approximately 11.8 dB. The receiver is actually only
degraded as predicted by effective C / N0 , but the
precorrelation C / N0 indicates much worse degradation.
Since effective C / N0 relates directly to receiver
performance in areas such as acquisition, carrier tracking,
and data demodulation, these results show that
precorrelation C / N0 does not report reliable information
38
36
34
32
30
-160
Effective C/N0
Precorrelation C/N0
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 2. Theoretical Results for Narrowband Interference
at Band Center, Receiver Front-End Bandwidth 16 MHz
precorrelation C / N0 for a receiver with 16 MHz front-end
bandwidth. While precorrelation C / N0 understates the
effect of interference near band center, it overstates the
effect of interference near band edge.
46
44
42
40
C/N0 (dB-Hz)
Consider next narrowband interference offset from center
frequency by 3.5 times the chip rate (3.5805 MHz for C/A
code). Figure 3 compares effective C / N0 and
precorrelation C / N0 for a receiver with 8 MHz front-end
bandwidth. In contrast to Figures 1 and 2, the precorrelation
C / N0 indicates that this interference degrades receiver
performance much worse than it actually does. As predicted
analytically in the previous section, precorrelation C / N0
overestimates the effect of the interference by
approximately 12 dB. For example, while the precorrelation
C / N0 reports significant degradation at interference power
of –125 dBW, the actual degradation of effective C / N0 is
quite mild.
38
36
34
46
32
Effective C/N0
Precorrelation C/N0
44
30
-160
42
C/N0 (dB-Hz)
38
36
34
32
Effective C/N0
Precorrelation C/N0
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 3. Theoretical Results for Narrowband Interference
Offset from Band Center by 3.5 Times the Chip Rate,
Receiver Front-End Bandwidth 8 MHz
Figure 4 shows the same quantities as in Figure 3 for a
receiver with 16 MHz front-end bandwidth. For this widerbandwidth receiver, precorrelation C / N0 underestimates
the effect of the interference by approximately 9 dB. The
effective C / N0 shows that this interference has almost the
same effect for front-end bandwidths of 8 MHz and
16 MHz, but the precorrelation C / N0 changes by 3 dB.
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
46
44
42
40
C/N0 (dB-Hz)
44
42
40
C/N0 (dB-Hz)
-145
Figure 6 compares effective C / N0 and precorrelation
C / N0 for a receiver with 8 MHz front-end bandwidth, and
interference whose spectrum is centered at band center,
with constant interference power corresponding to Cι / N0
of 75 dB-Hz and varying bandwidth. The effective C / N0
shows that the receiver’s ability to acquire, track carrier,
and demodulate data is degraded most when the
interference bandwidth is small, and that the same power
interference has nearly 8 dB less effect when the its
bandwidth is the same as the front-end bandwidth.
Precorrelation C / N0 does not indicate the greater
degradation caused by narrowband interference at band
center, but indicates that C / N0 is better than it actually is.
46
38
36
34
38
32
36
30
0
34
32
30
-160
-150
Figure 5. Theoretical Results for Narrowband Interference
Offset from Band Center by 7.5 Times the Chip Rate,
Receiver Front-End Bandwidth 16 MHz
40
30
-160
-155
Effective C/N0
Precorrelation C/N0
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 4. Theoretical Results for Narrowband Interference
Offset from Band Center by 3.5 Times the Chip Rate,
Receiver Front-End Bandwidth 16 MHz
Consider next narrowband interference offset from center
frequency by 7.5 times the chip rate (7.6725 MHz for C/A
code). Figure 5 compares effective C / N0 and
Effective C/N0
Precorrelation C/N0
1
2
3
4
5
Complex Bandwidth of Interference (MHz)
6
7
Figure 6. Theoretical Results for Interference Whose
Spectrum is Centered in the Band, with Cι / N0 of
75 dB-Hz, Receiver Front-End Bandwidth 8 MHz
Figure 7 shows the same results under the same conditions
as Figure 6, except with receiver front-end bandwidth of
16 MHz. Here again, the precorrelation C / N0 does not
report the greater degradation caused by narrowband
interference at band center, and instead reports less
degradation than actually occurs.
8
46
44
44
42
42
40
40
C/N0 (dB-Hz)
C/N0 (dB-Hz)
46
38
36
34
36
34
32
30
0
38
32
Effective C/N0
Precorrelation C/N0
2
4
6
8
10
Complex Bandwidth of Interference (MHz)
12
14
30
0
16
Effective C/N0
Precorrelation C/N0
1
2
3
4
5
Center Frequency of Interference (MHz)
6
7
8
Figure 7. Theoretical Results for Interference Whose
Spectrum is Centered in the Band, with Cι / N0 of 75 dBHz, Receiver Front-End Bandwidth 16 MHz
Figure 9. Theoretical Results for Narrowband Interference
with Cι / N0 of 75 dB-Hz, Receiver Front-End Bandwidth
16 MHz
Figure 8 compares effective C / N0 and precorrelation
C / N0 for a receiver with 8 MHz front-end bandwidth, and
interference with 10 kHz bandwidth, having a constant
interference power corresponding to Cι / N0 of 75 dB-Hz,
but varying center frequencies. The effective C / N0 shows
that the receiver’s ability to acquire, track carrier, and
demodulate data is degraded most when the interference is
concentrated near band center, and that the same power
interference has up to 15 dB less effect when the
interference bandwidth is far from band center, but still
within the front-end bandwidth. The precorrelation C / N0
does not report the greater degradation caused by
narrowband interference at band center. Instead it reports
less degradation than actually occurs for interference near
band center, and greater degradation than actually occurs
for interference far from band center.
Figure 10 compares predicted effective C / N0 and
precorrelation C / N0 for a receiver with 8 MHz front-end
bandwidth, and partial-band interference near band edge, at
different levels of received interference power. Also shown
are C / N0 values reported by a commercial receiver
referred to as receiver Model A. To apply the theory, the
receiver’s front-end bandwidth, the received signal power,
and the thermal noise level at the receiver must all be
known. Measurements using the methodology described in
[1] show that the front-end bandwidth of this receiver is
approximately 8 MHz, and that the receiver reports C / N0
based on postcorrelation measurement of the noise density.
The measurements used a received signal power level of
–157 dBW and noise density of –200 dBW/Hz. The
theoretical curves in Figure 10 show that precorrelation
C / N0 indicates degradation at much lower power levels
than actually disturb the effective C / N0 . In fact, effective
C / N0 is degraded to 40 dB-Hz only when the received
interference power exceeds –111 dBW. The C / N0 values
reported by the receiver closely match those theoretically
predicted for effective C / N0 , substantiating the theory and
the indications that the receiver reports C / N0 based on
postcorrelation measurement of the noise density.
46
44
42
38
36
46
34
44
32
30
0
42
Effective C/N0
Precorrelation C/N0
0.5
1
1.5
2
2.5
Center Frequency of Interference (MHz)
3
3.5
Figure 8. Theoretical Results for Narrowband Interference
with Cι / N0 of 75 dB-Hz, Receiver Front-End Bandwidth
8 MHz
Figure 9 shows the same results under the same conditions
as Figure 8, except with receiver front-end bandwidth of
16 MHz. Precorrelation C / N0 reports less degradation
than actually occurs for interference near band center, and
greater degradation than actually occurs for interference far
from band center.
4
40
C/N0 (dB-Hz)
C/N0 (dB-Hz)
40
38
36
34
Theoretical Effective C/N0
Theoretical Precorrelation C/N0
C/N0 Reported by Receiver
32
30
-160
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 10. Results for Band-Edge Interference, Theory
Uses Receiver Front-End Bandwidth 8 MHz, Model A
Receiver
46
44
42
40
C/N0 (dB-Hz)
To examine the sensitivity of the results in Figure 10 to
receiver front-end bandwidth, Figure 11 shows the same
results as in Figure 10, only with the theory evaluated
assuming a receiver front-end bandwidth of 16 MHz rather
than the 8 MHz that was measured. As expected, the
theoretical effective C / N0 changes little, and still matches
the measured data closely. The reported C / N0 does not
match the precorrelation C / N0 , which reports much
greater degradation than actually occurs.
38
36
34
46
Theoretical Effective C/N0
Theoretical Precorrelation C/N0
C/N0 Reported by Receiver
32
44
30
-160
-155
-150
-145
42
38
36
34
-120
-115
-110
Theoretical Effective C/N0
Theoretical Precorrelation C/N0
C/N0 Reported by Receiver
32
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 11. Results for Band-Edge Interference, Theory
Uses Receiver Front-End Bandwidth 16 MHz, Model A
Receiver
Figure 12 compares effective C / N0 and precorrelation
C / N0 for a receiver with 16 MHz front-end bandwidth,
and band-edge interference at different levels of received
interference power. Also shown are C / N0 values reported
by a commercially available receiver referred to as
Model B. Measurements using the approach described in
[1] show that the front-end bandwidth of this receiver is
approximately 16 MHz, and that the receiver reports
C / N0 based on precorrelation measurement of the noise
density. The measurements used a received signal power
level of –155 dBW and noise density of –200 dBW/Hz. The
C / N0 values reported by the receiver closely match those
theoretically predicted for precorrelation C / N0 ,
substantiating the theory and the indications that the
receiver reports C / N0 based on precorrelation
measurement of the noise density. These results show that
the receiver reports much greater degradation than actually
occurs for interference near band edge; other measurements
have shown that this receiver reports much less degradation
than actually occurs for interference near band center.
Theory shows that precorrelation C / N0 is very sensitive to
the front-end bandwidth used, especially when the
interference power is concentrated at the filter rolloff, as
occurs here. Figure 13 shows that using 18 MHz front-end
bandwidth produces theoretical predictions of
precorrelation C / N0 that closely match the experimental
results. An alternative approach would be to extend the
theory to model the actual filter rolloff rather than
approximating it as a rectangle.
46
44
42
40
C/N0 (dB-Hz)
C/N0 (dB-Hz)
-125
Figure 12. Results for Band-Edge Interference, Theory
Uses Receiver Front-End Bandwidth 16 MHz, Model B
Receiver
40
30
-160
-140
-135
-130
Total Interference Power (dBW)
38
36
34
Theoretical Effective C/N0
Theoretical Precorrelation C/N0
C/N0 Reported by Receiver
32
30
-160
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 13. Results for Band-Edge Interference, Theory
Uses Receiver Front-End Bandwidth 18 MHz, Model B
Receiver
To examine the sensitivity of the results in Figure 12 and
Figure 13 to receiver front-end bandwidth, Figure 14 shows
the same results as in these figures, only with the theory
evaluated assuming a receiver front-end bandwidth of
8 MHz rather than the 16 MHz that was measured in [1]. As
expected, the theoretical effective C / N0 shows very little
change with this change. The data does not match the
precorrelation C / N0 , demonstrating that this quantity is
very sensitive to front-end bandwidth, unlike the effective
C / N0 that actually describes the degradation.
46
44
42
C/N0 (dB-Hz)
40
38
36
34
Theoretical Effective C/N0
Theoretical Precorrelation C/N0
C/N0 Reported by Receiver
32
30
-160
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 14. Results for Band-Edge Interference, Theory
Uses Receiver Front-End Bandwidth 8 MHz, Model B
Receiver
Figure 15 compares effective C / N0 and precorrelation
C / N0 for a receiver with 24 MHz front-end bandwidth,
and band-edge interference at different levels of received
interference power. The effective C / N0 changes very little
with the difference in receiver front-end bandwidth, while
the precorrelation C / N0 has greater discrepancy. A
receiver that reports precorrelation C / N0 would report
degradation at almost 15 dB lower interference power than
degradation actually occurs.
46
44
42
C/N0 (dB-Hz)
40
38
36
34
32
30
-160
Effective C/N0
Precorrelation C/N0
-155
-150
-145
-140
-135
-130
Total Interference Power (dBW)
-125
-120
-115
-110
Figure 15. Theoretical Results for Band-Edge Interference,
Receiver Front-End Bandwidth 24 MHz
CONCLUSIONS
Theory and experimental results have been presented to
consider some ways that receivers estimate the effect of
interference on C / N0 , under the assumptions that the
interference is continuous, Gaussian, stationary, and
circularly symmetric. Also, the results here assume that the
receiver front end responds linearly to the noise and
interference. At some point, the interference power can be
high enough so that functions like mixing and quantization
are affected in ways not considered here. Nonetheless, this
theory appears to apply for the receivers considered here.
Effective C / N0 is not an arbitrary descriptor of signal
quality, but instead is directly and uniquely related to the
performance of essential receiver functions such as
acquisition, carrier tracking, and data demodulation.
Some GPS receiver designs may attempt to estimate C / N0
from precorrelation measurements. While this estimation
approach can be reliable in noise and interference that is flat
over the front-end bandwidth, it is unreliable for
interference whose spectrum is not flat over that bandwidth.
When the interference power is concentrated near band
center, the precorrelation C / N0 is higher than the effective
C / N0 , so that the reported C / N0 does not indicate
receiver degradation that actually occurs. Conversely, when
the interference power is concentrated near band edge, the
estimated C / N0 from the precorrelation C / N0 is lower
than the true effective C / N0 , so the reported C / N0
indicates that the receiver is degraded more than it actually
is. Further, precorrelation C / N0 is very sensitive to the
receiver front-end bandwidth, making it difficult to interpret
and to predict theoretically.
The consequences of reporting precorrelation C / N0
depend on how the receiver uses this erroneous
information. If the information is merely reported, then the
reliability of the reported C / N0 may only be misleading,
and have little other consequence. Conversely, if the
receiver adjusts its state or mode based on the reported
C / N0 (e.g., it does not track a signal with too low a
reported C / N0 ), then the consequences of erroneously
estimating C / N0 can be more significant. In particular, the
receiver may be unreliable when exposed to interference
that does not have a flat spectrum.
Furthermore, using reported C / N0 is not a reliable way to
assess the effect of interference on a receiver. Unless it is
determined that the receiver uses reliable algorithms for
reporting the effect of interference (i.e., that the receiver
reliably reports the effective C / N0 ), curious results may
be obtained, yielding misleading conclusions.
Based on this work, it is recommended that receivers report
C / N0 based on postcorrelation quantities, or an equivalent
technique that approximates effective C / N0 . Further,
C / N0 reported by receivers should be used only after
determination that the reported C / N0 is a reliable
indicator of receiver performance—that it corresponds to
effective C / N0 .
ACKNOWLEDGMENTS
This work was supported by Air Force contract F19628-00C-001. Thanks to Karl Kovach for initially pointing out the
curious behavior analyzed in this paper, and to Joseph Leva
and Shawn Yoder for performing measurements reported in
this paper.
REFERENCES
1.
2.
J. T. Ross, et al., “Effect of Partial-Band Interference
on Receiver Estimation of C/N0: Measurements,”
Proceedings of ION 2001 National Technical Meeting,
Institute of Navigation, January 2001.
J. W. Betz, “Effect of Narrowband Interference on GPS
Code Tracking Accuracy,” Proceedings of ION 2000
National Technical Meeting, Institute of Navigation,
January 2000.
3.
4.
5.
K. R. Kolodziejski and J. W. Betz, Effect of Non-White
Gaussian Interference on GPS Code Tracking
Accuracy, The MITRE Corporation Technical Report
MTR99B21R1, June 1999.
J. W. Betz and K. R. Kolodziejski, “Generalized
Theory of GPS Code Tracking Accuracy with an
Early-Late Discriminator,” to be Submitted to IEEE
Transactions on Aerospace and Electronic Systems.
C. Cahn, “Description of the SiRF Technology
GPS/WAAS Receiver,” Informal Communication,
February 2000.